Find the antiderivative F of f(x) = 4x³ + 2 sin x that satisfies F(0) = 1.

1. 1 · Write what is given and what is to be found. Given: F′(x) = 4x³ + 2 sin x and F(0) = 1. To find: F(x).
2. 2 · Write the formula or rule. F(x) = ∫ f(x) dx. Use ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C and ∫ sin x dx = −cos x + C.
3. 3 · Integrate term by term. F(x) = 4 × x⁴/4 + 2(−cos x) + C = x⁴ − 2 cos x + C.
4. 4 · Use the given condition to find C.
   • F(0) = 0 − 2 cos 0 + C = −2 + C
   • −2 + C = 1, so C = 3
5. 5 · Check and box the answer. F(x) = x⁴ − 2 cos x + 3. Check: F′(x) = 4x³ + 2 sin x and F(0) = −2 + 3 = 1.

Scoring tip: Find C before you box the answer. Leaving + C in a question that gives a point loses a mark.